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Unformatted text preview: Exercise A monopolist faces the demand curve Q = 120 - 3P. Find the revenue maximizing price and quantity: Maximizing revenue is equivalent to maximizing profit when MC = 0. By the (P-choke + MC)/2 rule, the optimal price is 20. Then, the optimal quantity is 60. Exercise Find the profits at the Nash equilibrium. Consider two firms that play a simultaneous move Cournot competition game. The market inverse demand is given by P = 120 Q. The marginal cost is zero for both firms: P1 = 120 Q1 Q2 ... MR1 = 120 Q1 2(Q2) = 0 ... solve for Q1, then set Q1 and Q2 = Qne, solve ... obtain Qne and Pne, solve for profits. Profits = (Pne MC) x Qne Exercise Consider now a Cournot game where each firm is allowed to choose either the Nash equilibrium quantity or half the monopoly quantity, only. Draw the normal form of the simultaneous move game. By the P-choke/MC rule the monopoly profitmaximizing price is 60, which makes the monopoly quantity 60. If each firm produces half of that, each makes a profit of 1800. If one firm produces half the monopoly quantity and the other produces the Nash equilibrium outcome, the price is 50: (P = 120 Qnash .5Qmonop) The profits are 1500 and 2000, respectively: Nash= (50-0)(40) .5 Monop= (50-0)(30) Nash Half Monopoly Nash 1600, 1600 1500, 2000 Half Monopoly 2000, 1500 1800, 1800 Exercise Argue why the game above is a prisoner's dilemma: Each firm has a dominant strategy. Also, the dominant strategy equilibrium is pareto inferior to (1800, 1800). Exercise A monopolist produces at constant marginal cost and faces two segmented markets with the following demand curves: Qa = 100- P ... Qb = 100 2(P) In which market will he charge a higher price if he maximizes profits? : E = P/(Pchoke - P) The first demand has a higher choke price and is therefore more inelastic for any price. By the mark up elasticity formula: the lower the elasticity, the higher the price you can charge, w/o the quantity changing that much. Therefore, the price should be higher in segment A. Exercise Assume that you estimate the elasticity of demand for luxury cars to be 0.8. On the basis of this information, can you conclude that at least one car firm is not maximizing profits? Explain: It is true that any firm that maximizes profits will produce at the elastic (E > 1) part of the demand. However, the estimate of 0.8 refers to the industry demand...isn't informative about the pricing behavior of individual firms. Exercise Consider the Stackelberg (sequential) version of a two-player Cournot competition game. Would you rather be the follower, or the leader? (Assume that both firms have the same constant marginal cost.): We know that in the Stackelberg version of the Cournot game, the following statements hold, relative to the Nash equilibrium outcome of the simultaneous move game: The leader wants the follower to decrease his quantity, because the game has negative externalities. The leader achieves this by increasing his quantity, because actions are strategic substitutes. This lowers the profits of the follower, because the game has negative externalities. The profit of the leader goes up in all games. Therefore, the leader has higher profits than the follower and you would like to be the leader. Exercise Consider a market with positive network externalities. Are the actions of consumers strategic substitutes or strategic complements? Explain: If other consumers are more likely to buy, I am also more likely to buy, because with positive externalities my valuation increases with the number of people that use a good or a service. Exercise Suppose you are an antitrust regulator. Argue why you would have a basis for supporting a merger between a gin firm and a tonic firm: Gin and tonic are complement goods and the simultaneous move price game has negative externalities. When the two firms merge, they internalize this externality and they decrease their prices, relative to the Nash equilibrium outcome. Their joint profits increase and consumers benefit from lower prices. Exercise A publishing house produces a paperback and a hardcover edition of the same book. The marginal costs are 10 and 18 respectively. The prices are 20 and 24 respectively. The firm has also calculated that at these prices the elasticities of demand are 2 and 4 respectively. Can you tell whether the firm is maximizing profits? Explain why, or why not. Use formula: MC = P(1 (1/E))...each firm maxes own profits, but not joint profits (they play a NE, which isn't necessarily maxing joint profits) Exercise Three firms play a Cournot quantity competition game. The market inverse demand is P = 100 Q and each firm has a constant marginal cost of 10. The timing of the game is as follows. First, firm 1 chooses its quantity. Then, firms 2 and 3 observe the quantity of firm 1 and they choose their quantities simultaneously. Assuming that firms 2 and 3 will play according to the Nash equilibrium, what is the quantity that firm 1 should choose to maximize its profits? Assume that firm 1 chooses quantity x. Then, firms 2 and 3 play a standard Cournot game with market inverse demand P =100 x Q ... MR2 = 100 x Q3 2(Q2) = 10 ... Q2 = (90 x Q3) / 2 ... plug in Q2 for Q3 ... Qne = (90 x) / 3 ... firm 1 knows this, via backwards induction: plug into original equation, solve for P, find MR with new equation, set equal to MC, solve. REMEMBER: there are 2 firms, so the equation is: P=100 x - 2(Qne) Profit maximization, marginal cost and marginal revenue 1. A firm maximizes profits at the point where MC = MR. 2. In monopoly MR < P. In fact, MR can also be negative. 3. The previous point implies that the MR curve lies below the inverse demand. 4. For a linear demand, the MR curve has the same intercept and two times the slope as the inverse demand. 5. In general, MR = P[1-1/ ] The monopoly pricing problem In the typical problem, you are given the demand and the marginal cost curves. One needs to follow the following steps: 1. Obtain the inverse demand. 2. Obtain the MR curve. For linear demand curves remember the same-intercepttwice-the-slope rule. 3. Equate MR to MC to obtain the profit maximizing quantity. 4. Plug in the profit maximizing quantity into the inverse demand to obtain the profit maximizing price. Implications of the mark-up elasticity formula MC = P[1-1/]: 1. A monopolist always produces in the elastic portion of his demand. 2. A monopolist charges a higher price when demand becomes more inelastic. 3. MR is negative when < 1. MR is positive when > 1. Revenue is maximized when = 1. 4. In the monopoly model there is no supply curve. Price depends on elasticity and marginal cost. As a result, the effect of taxes and exogenous demand shifts on price are ambiguous. Perfect Competition 1. Perfect competition is a special case of Pricing with Market Power. As the market becomes more and more competitive, the demand curve that a firm faces becomes more and more flat. In the extreme case, the demand curve is horizontal and its elasticity is infinite. Then, by the formula MR = P [1 1/E], it is immediate to see that MR = P. 2. The only difference from the monopoly model is that, in perfect competition, the MC curve (above min AC) is the individual supply curve. 3. The sum of all individual supply curves is the market supply curve. 4. The market demand curve is NOT the sum of the individual demand curves. 5. Make sure you understand the mechanics of the "dual graph" (firm's decision on the left graph and the market on the right graph). Price competition (or Bertrand competition) In a typical problem you need to know how to execute the following steps: 1. Derive the reaction curves. You can easily do this using the usual techniques from monopoly pricing, i.e. Pchoke-plus-MC-divided-by-2 rule or equating MC to MR. 2. Graph the reaction curves and compute the NE, i.e. the point where the two reaction curves intersect. 3. Calculate profits at the NE. 4. Compute the Stackelberg outcome, by plugging the Follower's reaction curve into the Leader's demand curve. 5. You should be able to explain how the NE prices and profits compare to the Stackelberg ones. To do this, you need to use the concepts of positive vs. negative externalities. 6. Do firms have an incentive to collude? If they collude, will prices be higher or lower? (answer depends crucially on positive or negative externalities.) Quantity Competition (or Cournot competition) Same as above, but now the choice variables are quantities, not prices. Also, notice that now the reaction curves are downward sloping and the game has negative externalities. Price Competition as a sequential game (Stackleberg). Firm 1 leads, 2 follows. 1. Use reaction curve of firm 2 in firm 1's demand. 2. Find profit maximizing price for firm 1 (i.e. P1). 3. Use firm 2's reaction curve to find P2. 4. Calculate profits. Exercise The city of New Haven has estimated that the number of parking tickets, T, that it collects each month depends on the fine, F, according to the following relationship: T = 240 2F. Assuming that each ticket has an administrative cost of 10 dollars, what is the profit maximizing fine?: The inverse demand is F = 120 1/2T. The profit-maximizing fine (price) is the midpoint of F-choke and marginal cost. Hence, the profit maximizing fine is 1/2(120+10) = 65 Exercise Suppose that Microsoft's wholesale price of Windows XP is $140 and that it has correctly calculated that this price maximizes profit. Suppose also that Microsoft's marginal cost for copies of XP is $20. What is Microsoft's elasticity of demand at the price of $140?: The mark-up elasticity formula implies that: 20 = 140(1-1/E) ... Straightforward algebra shows that E = 7/6 Exercise Are the following statements true or false? Explain. 1. In a Nash equilibrium at least one player plays a dominant strategy. FALSE: The battle of the sexes game is a counter-example for the first claim. 2. A Nash equilibrium is always pareto efficient. FALSE: The prisoner's dilemma is a counter-example to the second claim. Exercise Three firms play a price competition game. They face constant marginal cost of 10 and their demand curves are given by: Q1 = 80 2P1 + P2 + P3, Q2 = 80 2P2 + P1 + P3, Q3 = 80 2P3 + P1 + P2 ... Find the Nash equilibrium prices. Hint: The game is symmetric: Derive the reaction curve for firm 1 as a function of the prices of firms 2 and 3. To do this, use the mid-point pricing rule. Then, solve for the NE prices, which you know they have to be equal, because the game is symmetric. Solve for P1. Then: .5(P1 + MC). Then: set all "P's" to Pne. Solve. Exercise Consider a two-player game with numerical actions with positive or negative externalities. Let the players be 1 and 2. Suppose that there is a unique Nash equilibrium (A1N, A2N), which results in the payoffs (u1N, u2N). Consider in comparison the Stackleberg game in which player 1 is the leader. Denote the Stackleberg outcome by (A1S, A2S) and the associated payoffs by (u1S, u2S). You are told that A2N >A2S and that reaction curves have a positive slope. On the basis of this information, can you tell whether u2N <u2S or u2N >u2S? The following table may come useful. What does L want F to do? A2 Falls Negative Externalities How does L achieve this? A1 Falls Reaction curves have positive slope How does L's action affect F's payoff? U2 Rises Negative Externalities My notes Explicit market segmentation: MRa(Qa) = MRb(Qb) = MC(Qa+Qb) quantity supplied doesn't change: switch units around Monopolies: price determined by elasticity and MC (taxes and demand shifts have ambiguous affects) : MR < P (MR can be negative): E<1: MR neg E=1: max. rev. E>1: MR pos Cournot: Pnash = (Pchoke + 2(MC))/2 MR = MC = P(1 (1/E)) ... perfect competition: E >> infinity, thus MR = P P = (Pchoke+MC)/2 ... E = P/(Pchoke P) Min AC = MC, find q, plug in for ne, solve for P + externs: A increases price, B's profits increase...+ slope: if you want to increase other's price, increase your price (substitute goods) - externs: A decreases volume, B's volume decreases (complementary goods, too) Price competition as sequential game (Stackleberg, F1 is leader, F2 is follower): Use reaction curve of firm 2 in firm 1's demand. Find profit-maxing price for firm 1 (P1). Use firm 2's reaction curve to find P2. Calculate profits. Prisoner's dilemma: each player has incentive to cheat at other's expense. When both cheat, they are worse off than if neither did. Each has a dominant strategy. Consumer surplus = area under inverse demand and above price ...
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This note was uploaded on 12/02/2007 for the course ECON 110 taught by Professor Donaldbrown during the Spring '06 term at Yale.
- Spring '06